There are many good guides to Burns country. The first reference point is the Burns Encyclopaedia. For contemporary accounts of each parish community in Scotland in Burns's time, see:
Robert Donald Thornton, The Reading of Robert Burns as Reflected in his Letters and Poems (Unpublished BA honors thesis, Wesleyan University, CT, 1938).
John S. Robotham, “The Reading of Robert Burns,” Bulletin of the New York Public Library, 74:9 (November 1970): 561-576; reprinted in Carol McGuirk, Critical Essays on Robert Burns (1998), 281-297; in addition to a list of books, Robotham gives citations to document Burns's reading, knowledge, possession, or borrowing of each title.
J. Walter McGinty, Robert Burns the Book Lover (Kilkerran: Humming Earth, 2013).
Robert Crawford, The Bard: Robert Burns, A Biography (Princeton: Princeton Univ. Press; London: Jonathan Cape, 2009; revd ed. London: Pimlico, 2010): PR 4302 .C72 2009
James A. Mackay, RB: A Biography of Robert Burns (Edinburgh: Mainstream, 1992; repr. Alloway: Alloway Publishing, 2004): PR4331 .M32 1992
Ian McIntyre, Dirt & Deity: A Life of Robert Burns (London: HarperCollins, 1995; revd. ed. London: Constable, 2009): PR4331 .M35 1995
David Daiches, Robert Burns and His World (London: Thames and Hudson, 1971): PR4331 .D28 1971
Franklin Bliss Snyder, The Life of Robert Burns (New York: The Macmillan Company, 1932; repr. New York: Archon 1968): PR4331 .S6
Catherine Carswell, The Life of Robert Burns (London: Chatto and Windus, 1930; Edinburgh: Canongate Classics, 1990): PR4331 .C45 1990
Patrick Scott, ed., Robert Burns: A Documentary Volume [Dictionary of Literary Biography 383] (2018): 300 documents or extracts, 100 illustrations, over forty manuscript facsimiles, maps etc.
Nigel Leask, ed., Commonplace books, Tour Journals and Miscellaneous Prose, The Oxford Edition of the Works of Robert Burns, vol. 1 (Oxford: Oxford University Press, 2014); an annotated scholarly edition of Burns's writing other than poems and letters, arranged chronologically
George Scott Wilkie, Robert Burns: A Life in Letters (Glasgow: Neil Wilson Publishing, 2011).
James Currie, ed., The Works of Robert Burns: with an Account of His Life, and Criticisms on his writings, 4 vols. (1800), vol. 1: PR4300 1800 .L3
R. H. Cromek, ed., Reliques of Robert Burns: Consisting chiefly of original Letters, Poems, and Critical Observations on Scottish Songs (1808): PR 4302 .C7 1808
John Gibson Lockhart, Life of Robert Burns (1828): PR4331 .L6 1828
Allan Cunningham, ed., The Works of Robert Burns; With his life, 8 vols. (1834): PR4300 1834 .L4
James Hogg, "Memoir of Burns," vol. 5 of The Works of Robert Burns, 5 vols. (1836), ed. the Ettrick Shepherd [James Hogg] and William Motherwell: PR4300 1836 .G3
Robert Chambers, ed., The Life and Works of Robert Burns, 4 vols. (1851): PR4300 1851 .L4
William Scott Douglas, editorial material in his The Works of Robert Burns, 6 vols [vols. 1-3 Poetry, vols. 4-6 Prose] (1877-79):
Robert Chambers and William Wallace, The Life and Works of Robert Burns, 4 vols. (1896): PR4300 1896 .E14
H.B. Anderson, in Annals of Medical History, 10 (March 1928), 47-58; revised as Robert Burns, his medical friends, attendants and biographer (New York: Paul B. Hoeber 1928)
A. Beveridge, ‘”Groaning under the miseries of a diseased nervous System”: Robert Burns and Melancholy," in Scottish Medicine and Culture, ed. David Shuttleton and Megan Coyer (Amsterdam: Rodopi, 2014), 145-171
W. Buchanan, and W. Kean, "‘Robert Burns’s illness revisited’," Scottish Medical Journal, 27:1 (January 1982), 75-88; revised as "Robert Burns final illness revisited," Burns Chronicle, (1991), 60-71
W. Buchanan, "Robert Burns’s Rheumatology," Burns Chronicle, (1996), 237-243
Sir James Crichton-Browne, Burns from a New Point of View (London: Hodder and Stoughton, 1926; 2nd ed , London: Hodge, 1937)
William Findlay, Robert Burns and the Medical Profession (Paisley: Gardner 1898)
J. Purves-Stewart, The Immortal Memory of Burns: A Medical Aspect (London: for the Author, 1935)
Nigel Leask, "Burns, Wordsworth and the Politics of Vernacular Poetry," in Peter de Bolla, Nigel Leask and David Simpson (eds),Land, Nation and Culture, 1740–1840 (Basingstoke: Palgrave Macmillan, 2005), pp. 202–22.
Leith Davis, "The Poetry of Nature and the Nature of Poetry: Robert Burns and William Wordsworth," in Acts of Union: Scotland and the Literary Negotiation of the British Nation 1707–1830 (Stanford: Stanford University Press, 1998), pp. 107–43.
Liam McIlvanney, "Robert Burns and the Calvinist Radical Tradition," History Workshop Journal 40 (1995), pp. 133–49.
Michael Morris, "Robert Burns, Slavery Freedom and Abolition, 1786-1800" in Scotland and the Caribbean, c. 1740-1833: Atlantic Archipelagos (New York and London: Routledge, 2015).
Eric J. Graham, Burns & the Sugar Plantocracy of Ayrshire Ayr: Ayrshire Archeological and Natural History Society, 2009.
Nigel Leask , "Burns and the Poetics of Abolition," in The Edinburgh Companion to Robert Burns, ed. Gerard Carruthers (Edinburgh: Edinburgh UP, 2009): 47-60.
Clark McGinn, “Burns and Slavery,” Burns Chronicle (Spring 2010), 8-24.
Murray Pittock, “Slavery as a political metaphor in Scotland and Ireland in the Age of Burns,” in Robert Burns and Transatlantic Culture, ed. Sharon Alker, Leith Davis, and Holly Faith Nelson (Farnham: Ashgate, 2012), 19-30.